Abstract
The visibility representation (VR for short) is a classical representation of plane graphs. It has various applications and has been extensively studied. A main focus of the study is to minimize the size of the VR. It is known that there exists a plane graph $G$ with $n$ vertices where any VR of $G$ requires a grid of size at least (2/3)n x((4/3)n3) (width x height). For upper bounds, it is known that every plane graph has a VR with grid size at most (2/3)n x (2n5), and a VR with grid size at most (n1) x (4/3)n. It has been an open problem to find a VR with both height and width simultaneously bounded away from the trivial upper bounds (namely with size at most c_h n x c_w n with c_h < 1 and c_w < 2$).
In this paper, we provide the first VR construction with this property. We prove that every plane graph of n vertices has a VR with height <= max{23/24 n + 2 Ceil(sqrt(n))+4, 11/12 n + 13} and width <= 23/12 n. The area (height x width) of our VR is larger than the area of some of previous results. However, bounding one dimension of the VR only requires finding a good storientation or a good dual s^*t^*orientation of G. On the other hand, to bound both dimensions of VR simultaneously, one must find a good $st$orientation and a good dual s^*t^*orientation at the same time, and thus is far more challenging. Since storientation is a very useful concept in other applications, this result may be of independent interests.
BibTeX  Entry
@InProceedings{wang_et_al:LIPIcs:2011:3006,
author = {JiunJie Wang and Xin He},
title = {{Compact Visibility Representation of Plane Graphs}},
booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) },
pages = {141152},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897255},
ISSN = {18688969},
year = {2011},
volume = {9},
editor = {Thomas Schwentick and Christoph D{\"u}rr},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2011/3006},
URN = {urn:nbn:de:0030drops30064},
doi = {10.4230/LIPIcs.STACS.2011.141},
annote = {Keywords: plane graph, plane triangulation, visibility representation, storientation}
}
Keywords: 

plane graph, plane triangulation, visibility representation, storientation 
Collection: 

28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011) 
Issue Date: 

2011 
Date of publication: 

11.03.2011 